scholarly journals Bayesian joint muographic and gravimetric inversion applied to volcanoes

2019 ◽  
Vol 218 (3) ◽  
pp. 2179-2194 ◽  
Author(s):  
Anne Barnoud ◽  
Valérie Cayol ◽  
Valentin Niess ◽  
Cristina Cârloganu ◽  
Peter Lelièvre ◽  
...  

SUMMARY Gravimetry is a technique widely used to image the structure of the Earth. However, inversions are ill-posed and the imaging power of the technique rapidly decreases with depth. To overcome this limitation, muography, a new imaging technique relying on high energy atmospheric muons, has recently been developed. Because muography only provides integrated densities above the detector from a limited number of observation points, inversions are also ill-posed. Previous studies have shown that joint muographic and gravimetric inversions better reconstruct the 3-D density structure of volcanic edifices than independent density inversions. These studies address the ill-posedness of the joint problem by regularizing the solution with respect to a prior density model. However, the obtained solutions depend on some hyperparameters, which are either determined relative to a single test case or rely on ad-hoc parameters. This can lead to inaccurate retrieved models, sometimes associated with artefacts linked to the muon data acquisition. In this study, we use a synthetic example based on the Puy de Dôme volcano to determine a robust method to obtain the resulting model closest to the synthetic model and devoid of acquisition artefacts. We choose a Bayesian approach to include an a priori density model and a smoothing by a Gaussian spatial correlation function relying on two hyperparameters: an a priori density standard deviation and an isotropic spatial correlation length. This approach has the advantage to provide a posteriori standard deviations on the resulting densities. Using our synthetic volcano, we investigate the most reliable criterion to determine the hyperparameters. Our results suggest that k-fold Cross-Validation Sum of Squares and the Leave One Out methods are more robust criteria than the classically used L-curves. The determined hyperparameters allow to overcome the artefacts linked to the data acquisition geometry, even when only a limited number of muon telescopes is available. We also illustrate the behaviour of the inversion in case of offsets in the a priori density or in the data and show that they lead to recognizable structures that help identify them.

2021 ◽  
Vol 8 ◽  
Author(s):  
Anne Barnoud ◽  
Valérie Cayol ◽  
Peter G. Lelièvre ◽  
Angélie Portal ◽  
Philippe Labazuy ◽  
...  

Imaging the internal structure of volcanoes helps highlighting magma pathways and monitoring potential structural weaknesses. We jointly invert gravimetric and muographic data to determine the most precise image of the 3D density structure of the Puy de Dôme volcano (Chaîne des Puys, France) ever obtained. With rock thickness of up to 1,600 m along the muon lines of sight, it is, to our knowledge, the largest volcano ever imaged by combining muography and gravimetry. The inversion of gravimetric data is an ill-posed problem with a non-unique solution and a sensitivity rapidly decreasing with depth. Muography has the potential to constrain the absolute density of the studied structures but the use of the method is limited by the possible number of acquisition view points, by the long acquisition duration and by the noise contained in the data. To take advantage of both types of data in a joint inversion scheme, we develop a robust method adapted to the specificities of both the gravimetric and muographic data. Our method is based on a Bayesian formalism. It includes a smoothing relying on two regularization parameters (an a priori density standard deviation and an isotropic correlation length) which are automatically determined using a leave one out criterion. This smoothing overcomes artifacts linked to the data acquisition geometry of each dataset. A possible constant density offset between both datasets is also determined by least-squares. The potential of the method is shown using the Puy de Dôme volcano as case study as high quality gravimetric and muographic data are both available. Our results show that the dome is dry and permeable. Thanks to the muographic data, we better delineate a trachytic dense core surrounded by a less dense talus.


Author(s):  
Tianxing Wang ◽  
Junfeng Yang ◽  
Hongchao Wang ◽  
Hongwei Yu ◽  
Zhengyang Sun ◽  
...  

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Rujia Li ◽  
Liangcai Cao

AbstractPhase retrieval seeks to reconstruct the phase from the measured intensity, which is an ill-posed problem. A phase retrieval problem can be solved with physical constraints by modulating the investigated complex wavefront. Orbital angular momentum has been recently employed as a type of reliable modulation. The topological charge l is robust during propagation when there is atmospheric turbulence. In this work, topological modulation is used to solve the phase retrieval problem. Topological modulation offers an effective dynamic range of intensity constraints for reconstruction. The maximum intensity value of the spectrum is reduced by a factor of 173 under topological modulation when l is 50. The phase is iteratively reconstructed without a priori knowledge. The stagnation problem during the iteration can be avoided using multiple topological modulations.


2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.


2021 ◽  
Vol 129 (1) ◽  
Author(s):  
Serge Kräutle ◽  
Jan Hodai ◽  
Peter Knabner

AbstractWe consider a macroscale model of transport and reaction of chemical species in a porous medium with a special focus on mineral precipitation–dissolution processes. In the literature, it is frequently proposed that the reaction rate should depend on the reactive mineral surface area, and so on the amount of mineral. We point out that a frequently used model is ill posed in the sense that it admits non-unique solutions. We investigate what consequences this non-uniqueness has on the numerical solution of the model. The main novelty in this article is our proposal of a certain substitution which removes the ill-posedness from the system and which leads to better numerical results than some “ad hoc methods.” We think that the proposed substitution is a rather elegant way to get rid of the non-uniqueness and the numerical difficulties and is much less technical than other ideas. As a proof of concept, we present some numerical tests and simulations for the new model.


2019 ◽  
Vol 27 (3) ◽  
pp. 317-340 ◽  
Author(s):  
Max Kontak ◽  
Volker Michel

Abstract In this work, we present the so-called Regularized Weak Functional Matching Pursuit (RWFMP) algorithm, which is a weak greedy algorithm for linear ill-posed inverse problems. In comparison to the Regularized Functional Matching Pursuit (RFMP), on which it is based, the RWFMP possesses an improved theoretical analysis including the guaranteed existence of the iterates, the convergence of the algorithm for inverse problems in infinite-dimensional Hilbert spaces, and a convergence rate, which is also valid for the particular case of the RFMP. Another improvement is the cancellation of the previously required and difficult to verify semi-frame condition. Furthermore, we provide an a-priori parameter choice rule for the RWFMP, which yields a convergent regularization. Finally, we will give a numerical example, which shows that the “weak” approach is also beneficial from the computational point of view. By applying an improved search strategy in the algorithm, which is motivated by the weak approach, we can save up to 90  of computation time in comparison to the RFMP, whereas the accuracy of the solution does not change as much.


2018 ◽  
Vol 84 (6) ◽  
Author(s):  
G. J. Wilkie

The effect of electrostatic microturbulence on fast particles rapidly decreases at high energy, but can be significant at moderate energy. Previous studies found that, in addition to changes in the energetic particle density, this results in non-trivial changes to the equilibrium velocity distribution. These effects have implications for plasma heating and the stability of Alfvén eigenmodes, but make multiscale simulations much more difficult without further approximations. Here, several related analytic model distribution functions are derived from first principles. A single dimensionless parameter characterizes the relative strength of turbulence relative to collisions, and this parameter appears as an exponent in the model distribution functions. Even the most simple of these models reproduces key features of the numerical phase-space transport solution and provides a useful a priori heuristic for determining how strong the effect of turbulence is on the redistribution of energetic particles in toroidal plasmas.


2022 ◽  
Vol 19 (3) ◽  
pp. 175-204
Author(s):  
I. E. Pris

The renowned British philosopher Timothy Williamson talks about his philosophical views and main lines of research. Williamson is a metaphysical realist in a broad sense. Fir him there are true or false answers to questions about all aspects of reality. Classical logic is a universal true theory. Knowledge-first epistemology is an alternative to the traditional belief-first epistemology. The former takes the concept of knowledge as a basic concept, explaining other epistemic concepts, including belief, in its terms, whereas the latter does the opposite. Knowledge, not truth, is the fundamental epistemic good. The Gettier problem and the skeptical problem that arise within traditional epistemology are ill posed and therefore cannot be solved. Hybrid epistemological theories do not satisfy the principles of simplicity and beauty and are refuted by counter-examples. Epistemic contextualism is problematic, and relativism violates the semantics of the phenomena being explained. Knowledge does not entail knowledge about knowledge. Knowledge-how is a kind of knowledge-that. The distinction between a priori and a posteriori is superficial, and there are no analytical truths. The concept of qualia is unhelpful for solving the problems related to consciousness. The so-called “hard problem” of consciousness points to an area of conceptual confusions in which we do not know how to reason properly. Speculative metaphysics is quite a respectable enterprise. But progress in metaphysics is not automatic; it requires the right methodology.


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